where the numerator comes from fermionic integration and the denominator from bosonic one. The operator \(\eth \)  is the piece in the quadratic action that “maps \(\phi \)  to \(\tilde{\psi }\) ”Thus, while \(\mathcal {Q}\)  maps within the same multiplet, \(\eth \)  maps between different multiplets. It was shown in [1]} that a double-complex arises in this way. and \(H^\bullet (\eth )\)  is its cohomology. We can now decompose \(H^\bullet (\eth )\)  into irreducible representations of \(L\) , labelled by \(\alpha \) , which appears with finite multiplicity \(m_\alpha \) . Note that for \(D\)  elliptic, \(\alpha \)  is always finite (remember \(X\)  is compact, i.e. \(\eth \)  is Fredholm) whereas for transversally elliptic \(\eth \) , \(\alpha \)  is in general infinite. Let \(w_\alpha (\epsilon )\)  denote the weights of representation \(\alpha \)  depending on the equivariant parameters \(\epsilon \) . Then we can write
\(\operatorname{sdet}^{1/2}|_{H^\bullet (\eth )}\mathcal {L}=\prod _\alpha w_\alpha (\epsilon )^{m_\alpha /2}.\) 
